Analogous to position, velocity, and acceleration in normal “linear” motion where the derivative of position gives velocity and the derivative of velocity gives acceleration, angular motion can be determined in much the same way. Instead of position, velocity, and acceleration, there is angle, angular velocity, and angular acceleration. One can be derived from the other by the same mathematical functions in that the derivative of angle gives angular velocity, and the derivative of angular velocity gives angular acceleration. It is also understood that the inverse function of taking the derivative is to integrate (or integration).
In an ideal world, a single sensor can be used for one of the variables such as for angular velocity and then arrive to angle or angular acceleration by just doing the appropriate derivative or integration function. Since the world is not ideal in this regard, there are factors that affect and significantly degrade the ability to derive one parameter from another one. Noise, sampling rate, dynamic range, offsets, temperature drifts, etc. (other drifts) can cause such significant degradation. These derivation techniques are used today in some very high cost applications such as precision mine surveying, space navigation, and missile guidance, but such techniques are overwhelmingly affected for use in low cost applications.
Traditional inertial navigation systems (INS) include a collection of technologies that either provide direct measures or perform mathematical operations on sensor data to yield static angle, velocity (or rate), and acceleration, for use in a navigation computer to derive position. In the most advanced systems, there are a collection of sensors that provide linear acceleration, angular rate, and bearing. Sensors fail to exist that provide very accurate angular acceleration measures and hence need to be derived from the derivative of an angular rate sensor (gyroscope). Because of this, the error in the gyroscope is accumulated through the navigation computations. Existing INS technology is not small enough, light enough and cheap enough to embed or work with subscriber portable radios such as cellular phones or other portable consumer products.